Lattice Spacing Dependence of the First Order Phase Transition for Dynamical Twisted Mass Fermions

TL;DR

This paper investigates how the first-order phase transition seen in lattice QCD with Wilson-related actions depends on the lattice spacing by employing Wilson twisted mass fermions with a Wilson plaquette gauge action at three $\beta$ values. A scaling analysis is performed using a reference point defined by $(r_0 m_{PS})^2 = 1.5$ and dimensionless ratios $R_O$ and a scaling variable $\sigma$ to compare observables across $\beta$. The results show the phase transition weakens as the lattice spacing decreases, and key ratios such as $m_{PS}^2$, $f_{\chi}^{PS}$, and $m_{PS}/m_V$ exhibit little scaling violation even with nonzero twisted mass; data including $\mu=0$ at a coarser $\beta$ align on the same scaling curve. The study suggests that near-continuum physics may be accessible at modest lattice spacings but highlights that the first-order transition remains a barrier to chiral simulations, motivating exploration of alternative gauge actions (e.g., DBW2 or tree-level Symanzik improved) to suppress the transition.

Abstract

Lattice QCD with Wilson fermions generically shows the phenomenon of a first order phase transition. We study the phase structure of lattice QCD using Wilson twisted mass fermions and the Wilson plaquette gauge action are used in a range of beta values where such a first order phase transition is observed. In particular, we investigate the dependence of the first order phase transition on the value of the lattice spacing. Using only data in one phase and neglecting possible problems arising from the phase transition we are able to perform a first scaling test for physical quantities using this action.

Figures (7)

• Figure 1: The plaquette expectation value $\langle P\rangle$ as a function of $1/(2\kappa)$ at the three values of $\beta$ we have simulated. We also indicate the values of $a\mu$ which are scaled with $\beta$ such that $r_0\mu$ is roughly constant. The lines just connect the data points and only serve to guide the eye. For this study we have more simulations points than in the next figures and than in tables \ref{['tab:par5.1']}, \ref{['tab:par5.2']} and \ref{['tab:par5.3']}.
• Figure 2: In the graph on the left the PCAC quark mass is plotted as a function of $1/(2\kappa)$ at the three values of $\beta$ we have simulated. Positive values correspond to the low plaquette phase while negative values correspond to the high plaquette phase. The statistical errors are on this scale for most of the points smaller than the symbols. In the right plot we give a closeup of the $\beta=5.3$ results.
• Figure 3: The squared pion mass as a function of the PCAC quark mass at $\beta=5.3$.
• Figure 4: $r_0/a$ as a function of $1/(2\kappa)$ at $\beta=5.3$.
• Figure 5: The dependence of the pion mass on the quark mass, i.e. $R_{m_\mathrm{PS}^2}$ as a function of $\sigma$. Besides the data of this work, we added in the plot also results from Wilson fermion simulations at $\beta=5.6$Urbach:2005ji which were obtained at $\mu=0$.